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El. knyga: Fuzzy Graph Theory

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This book provides a timely overview of fuzzy graph theory, laying the foundation for future applications in a broad range of areas. It introduces readers to fundamental theories, such as Craine’s work on fuzzy interval graphs, fuzzy analogs of Marczewski’s theorem, and the Gilmore and Hoffman characterization. It also introduces them to the Fulkerson and Gross characterization and Menger’s theorem, the applications of which will be discussed in a forthcoming book by the same authors. This book also discusses in detail important concepts such as connectivity, distance and saturation in fuzzy graphs.

Thanks to the good balance between the basics of fuzzy graph theory and new findings obtained by the authors, the book offers an excellent reference guide for advanced undergraduate and graduate students in mathematics, engineering and computer science, and an inspiring read for all researchers interested in new developments in fuzzy logic and applied mathematics.

1 Fuzzy Sets and Relations
1(12)
1.1 Fuzzy Sets
1(4)
1.2 Fuzzy Relations
5(8)
2 Fuzzy Graphs
13(72)
2.1 Definitions and Basic Properties
13(2)
2.2 Connectivity in Fuzzy Graphs
15(5)
2.3 Forests and Trees
20(5)
2.4 Fuzzy Cut Sets
25(3)
2.5 Bridges, Cutsets, and Blocks
28(5)
2.6 Cycles and Trees
33(8)
2.7 Blocks in Fuzzy Graphs
41(9)
2.8 Strongest Strong Cycles and 0-Fuzzy Graphs
50(8)
2.9 Fuzzy Line Graphs
58(4)
2.10 Fuzzy Interval Graphs
62(14)
2.10.1 Fuzzy Intersection Graphs
63(1)
2.10.2 Fuzzy Interval Graphs
64(2)
2.10.3 The Fulkerson and Gross Characterization
66(2)
2.10.4 The Gilmore and Hoffman Characterization
68(8)
2.11 Operations on Fuzzy Graphs
76(9)
3 Connectivity in Fuzzy Graphs
85(42)
3.1 Strong Edges in Fuzzy Graphs
85(7)
3.2 Types of Edges in Fuzzy Graphs
92(7)
3.3 Vertex Connectivity and Edge Connectivity of Fuzzy Graphs
99(17)
3.4 Menger's Theorem for Fuzzy Graphs
116(11)
4 More on Blocks in Fuzzy Graphs
127(28)
4.1 Blocks of a Fuzzy Graph
127(7)
4.2 Critical Blocks and Block Graph of a Fuzzy Graph
134(5)
4.3 More on Blocks in Fuzzy Graphs
139(7)
4.4 Connectivity-Transitive and Cyclically-Transitive Fuzzy Graphs
146(9)
5 More on Connectivity and Distances
155(40)
5.1 Connectedness and Acyclic Level of Fuzzy Graphs
155(6)
5.2 Cycle Connectivity of Fuzzy Graphs
161(8)
5.3 Bonds and Cutbonds in Fuzzy Graphs
169(5)
5.4 Metrics in Fuzzy Graphs
174(9)
5.5 Detour Distance in Fuzzy Graphs
183(12)
6 Sequences, Saturation, Intervals and Gates in Fuzzy Graphs
195(36)
6.1 Special Sequences in Fuzzy Graphs
195(7)
6.2 Saturation in Fuzzy Graphs
202(12)
6.3 Intervals in Fuzzy Graphs
214(7)
6.4 Gates and Gated Sets in Fuzzy Graphs
221(10)
7 Interval-Valued Fuzzy Graphs
231(40)
7.1 Interval-Valued Fuzzy Sets
231(2)
7.2 Operations on Interval-Valued Fuzzy Graphs
233(8)
7.3 Isomorphisms of Interval-Valued Fuzzy Graphs
241(1)
7.4 Strong Interval-Valued Fuzzy Graphs
242(2)
7.5 Interval-Valued Fuzzy Line Graphs
244(5)
7.6 Balanced Interval-Valued Fuzzy Graphs
249(6)
7.7 Irregularity in Interval-Valued Fuzzy Graphs
255(6)
7.8 Self-centered Interval-Valued Fuzzy Graphs
261(10)
8 Bipolar Fuzzy Graphs
271(36)
8.1 Bipolar Fuzzy Sets
271(2)
8.2 Bipolar Fuzzy Graphs
273(8)
8.3 Isomorphisms of Bipolar Fuzzy Graphs
281(5)
8.4 Strong Bipolar Fuzzy Graphs
286(6)
8.5 Regular Bipolar Fuzzy Graphs
292(3)
8.6 Bipolar Fuzzy Line Graphs
295(5)
8.7 Connectivity in Bipolar Fuzzy Graphs
300(7)
Bibliography 307(8)
Index 315
Dr. Sunil Mathew is currently a Faculty Member in the Department of Mathematics, NIT Calicut, India. He has acquired his masters from St. Josephs College Devagiri, Calicut, and Ph.D. from National Institute of Technology Calicut in the area of Fuzzy Graph Theory. He has published more than 75 research papers and written two books. He is a member of several academic bodies and associations. He is editor and reviewer of several international journals. He has an experience of 20 years in teaching and research. His current research topics include fuzzy graph theory, bio-computational modeling, graph theory, fractal geometry, and chaos.

Dr. John N. Mordeson is Professor Emeritus of Mathematics at Creighton University. He received his B.S., M.S., and Ph.D. from Iowa State University. He is a Member of Phi Kappa Phi. He is the President of the Society for Mathematics of Uncertainty. He has published 15 books and 200 journal articles. He is on the editorial board of numerous journals. He has served as an external examiner of Ph.D. candidates from India, South Africa, Bulgaria, and Pakistan. He has refereed for numerous journals and granting agencies. He is particularly interested in applying mathematics of uncertainty to combat the problem of human trafficking.





Dr. Davender S. Malik is a Professor of Mathematics at Creighton University. He received his Ph.D. from Ohio University and has published more than 55 papers and 18 books on abstract algebra, applied mathematics, graph theory, fuzzy automata theory and languages, fuzzy logic and its applications, programming, data structures, and discrete mathematics.
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